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Conclusions & future research Take home points Collusion-resistance = truthfulness, when approximating OPT with lotteries for digital goods Lotteries much more expressive than universally truthful auctions New lower bounding technique based on Carver’s result about inconsistent systems of linear inequalities What next? Further applications/implications of Carver’s theorem? Lotteries for settings different than digital goods? E.g., goods with limited supply

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Extension to any finite domain For a domain of size d, we can easily get a d- approximation of OPT with a collusion-resistant lottery Straightforward generalization of the upper bound for {L,H} domain For any d and ε>0, there exist d values such that no truthful lottery can approximate OPT better than d-ε over the domain given by those values (Non-trivial) generalization of the lower bound for binary domains

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Lotteries for domains [1,H] Easy to come up with a collusion-resistant lottery guaranteeing a ln(H)+1 approximation of OPT Bidder bidding b i wins with probability ln(e*b i )/ln(e*H) Matching Lower bound can be proved for any truthful lottery over [1,H] Proof uses a technique designed for universally truthful auctions by [Goldberg et al, GEB 2006]… … And a bijection between truthful lotteries and universally truthful auctions E.g., the ½-approximating lottery for {L,H} can be viewed as a uniform distribution over two simple auctions charging H and L respectively

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Related literature Focus for “competitive auctions” [Goldberg et al, GEB 2006] is on F (2) rather than OPT, as OPT is “impossible” to approximate This research can then also be seen as the study of the implications of the knowledge of the domain on the approximation of OPT Lotteries are a more natural interpretation of universally truthful auctions In certain cases (ie, when Cumulative Distribution Functions do not have Probability density functions) lotteries are far more expressive than randomized auctions